In the eighties of the twentieth century, Benioff and Feynman proposed the concept of quantum computation, and then P.W. Shor gave the first quantum algorithm of very large integer factorization [1] in 1994 and L.K. Grover proposed a quantum algorithm which searches a marked state in an unordered list [2] in 1996. Quantum computation has been widely paid attention and become a challenging research line. Fuzzy logic, evolution calculation, and neural networks are regarded as the most promising three important aspects in the artificial intelligence field, which compose intelligence calculation (soft calculation) and have much comparability with quantum computation. Therefore, the combination of them would bring promising research in theory.

Some research results in the nineties of the twentieth century show that [3] - [5] the information processing of brain may be related to the quantum states; there may exists the effect of quantum mechanics in brain, and the quantum system has the same dynamics features as the biology neural networks. Fatt and Katz proved that in biology neuron, the neurotransmitter is released from the nerve terminals by the form of multi-molecule and small units, and they call this kind of small units quantum. Each of small units (quantum) describes the minimal unit that the neurotransmitter is released. In general, although the number of quantum in each synaptic response of neurons may be different, the number of molecules in each quantum is the same. Therefore, the combination of quantum computation and ANN may preferably simulate the information processing of brain.

However, as yet, there is little understanding of the essential components of artificial neural networks based on quantum theoretical concepts and techniques. The basal model and theory of quantum neural networks are in research. At present, there is not a set of perfect theory to direct the construction of model. Since Kak firstly proposed the concept of quantum neural computation [6] in 1995, the quantum neural networks have been paid attention, and many novel ideas and elementary model have been proposed. In 1997, N. B. Karayiannis et al. proposed the model of quantum neural networks with multilevel hidden neurons based on the superposition of quantum states in quantum theory. ZHU Da-qi et al. applied this model to fault diagnosis for photovoltaic radar electronic equipment, and acquired the satisfied results [7] . In 2000, Matsui et al. proposed a quantum neural networks model based on the single bit rotation gate and the 2-bit controlled-NOT gate, presented the algorithm of model, and investigated its performance on solving the 4-bit parity check problem and the function approximation problem [8] . In this model, the input is based on quantum bits (qubits). However, in many actual problems, the system input is the real vector in Hilbert space, and the method is not presented to transform the real to the quantum state in Ref. [8] . For the algorithm of model, the only two iterative equations in complex number are given based on the gradient descent algorithm. Neither the gradient computation formula nor the input-output relationship of networks is presented. Therefore, it is not easy for the reader to simulate the algorithm and apply this model to actual problem.

In this paper, based on the universality of the single qubit rotation gate and the two-qubit controlled-NOT gate, a quantum neuron model and three layers quantum back-propagation neural networks (QBP) model are proposed. The learning algorithm in Ref. [8] is improved. A transform method from the real to the qubit is proposed. On the basis of the probability amplitude vector of qubits, the input-output relationship of the model is presented. To facilitate the practical application, the learning algorithm is deducted in detail and the implementing scheme is given. The continuity of this model is theoretically proved. Two application examples are designed, and the simulation results show that this model and algorithm are evidently superior to the conventional back-propaga- tion networks (CBP) in three aspects: convergence speed, convergence rate, and robustness.

For the training samples described in the n-dimension Euclid-space, the transform formula to realize quantum state description of the training samples is defined as follows

where.

In the QBP described by Figure 2, there are three groups of parameters, the rotation parameter, the reversal parameter and the link-weights, that need updating. The error function is defined as follows

where and are the normalized desired output and the practical output, respectively.

Let, , Equation (12) can be rewritten as

Let, ,. According to the gradient descent algorithm,

The parameter updating rules are as follows

where is the learning coefficient.

To testify the validity of QBP, an actual example is designed and the QBP is compared with the CBP in this section. To make comparison equitable, the QBP adopts the same structure and parameters as CBP in the simulations.

The XOR problem has been the typical example of the neural networks learning algorithm research. This problem is the simplest example for nonlinear classification problem, which the corresponding optimization surface is irregular, and exists some local minimum. In this simulation, The XOR problem is generalized from four points to sixteen points. The point set is shown in Figure 3.

Firstly, we investigate how to change for the convergence rate when the learning coefficient changes. The networks structure is set to 2-10-1, the restriction error is 0.1, and the restriction iteration steps are 2000. The learning coefficient is from {0.1, 0.2, ・・・, 1.0}. This example is simulated 100 times for each learning coefficient by the QBP and the CBP, respectively. When the learning coefficient changes, the maximum of convergence rate of QBP is 87% and the minimum is 51%. However, the convergence rate of CBP changes in a large range when the learning coefficient changes. When the learning coefficient is less than 0.2, the convergence rate of CBP is 0%, and when the learning coefficient is more than 0.2, the maximum of convergence rate of CBP is

only 59%. The comparison result is shown in Figure 4.

The comparison result shows that the QBP is evidently superior to the CBP in both the convergence rate and the robustness.

Secondly, we investigate how to change for the iteration steps when the learning coefficient changes. The networks structure is set to 2-10-1 , the restriction error is 0.1, and the restriction iteration steps are 10,000. The learning coefficient is still from {0.1, 0.2, ・・・, 1.0}. This example is simulated 100 times for each learning coefficient by the QBP and the CBP, respectively. When the learning coefficient changes, for the average iteration steps of QBP, the maximum is 1838 steps, and the minimum is 151 steps; for the minimum iteration steps of QBP, the maximum is 336 steps, and the minimum is 27 steps. However, for the average iteration steps of CBP, the maximum is 7308 steps, and the minimum is 2021 steps; for the minimum iteration steps of CBP, the maximum is 4314 steps, and the minimum is 463 steps. The comparison result is shown in Figure 5.

The comparison result shows that the QBP is evidently superior to the CBP in both the iteration steps and its fluctuation range when the learning coefficient changes.

Finally, we investigate how to change for the iteration steps when the restriction error changes. The networks structure is set to 2-10-1 . It is known from Figure 4 and Figure 5 that the performance of QBP and CBP is the best when the learning coefficient is set 0.9. Therefore, the learning coefficient is set 0.9. The restriction error is from {0.10, 0.09, ・・・, 0.01}. This example is simulated 100 times for each restriction error by the QBP and the CBP, respectively. The comparison result is shown in Figure 6.

When the restriction error changes, for the average iteration steps of QBP, the maximum is 8575 steps, and the minimum is 431 steps; for the minimum iteration steps of QBP, the maximum is 1816 steps, and the minimum is 30 steps. However, for the average iteration steps of CBP, the maximum is 17,100 steps, and the minimum is 1304 steps; for the minimum iteration steps of CBP, the maximum is 14,525 steps, and the minimum is 659 steps. Hence, when the restriction error changes, the iteration steps and its fluctuation range of QBP is far less than that of CBP, which takes on the preferable robustness.

On the basis of the qubits and the universal quantum gates, a quantum BP neural networks model is proposed; the learning algorithm of this model is designed; and the continuity of this model is proved. The simulation result shows that this model and algorithm are superior to conventional BP networks in three aspects: the convergence speed, convergence rate, and robustness, by an actual example of pattern recognition and function approximation.

Jianping Li, (2015) Quantum-Inspired Neural Networks with Application. Open Journal of Applied Sciences,05,233-239. doi: 10.4236/ojapps.2015.56024